Mechanical Design

Ben Potsaid

Control System Design

February 26, 2003

Mechanical Design

Lecture Contents

•Miscellaneous Components•Gears and Belts •Flexible Couplings

•Bearings •Fixing Components to Shafts

•Dynamics

•Stress and Strain Analysis

•Heat Transfer Analysis

•PDE &The Finite Element Method

•Design Studies•Flexible Mirror Positioning (Robust Design)

•Pendubot Link Design (Mass & Stress Optimized)

•Robot Link Design (Controllability Optimized)Movies

Mechanical Design

Gears and Belts

Instantaneously, gears and pulleys act like levers

Mechanical Design

Gears and Belts (2)

Conservative System

(Power in = Power out)

2

1

1

2

2211

ττ

θθ

θτθτθτ

==

=

=

!!!!!

N

Power

21

2

1

2

1

ττττ

NrrN

=

==

Thermodynamic(conservation of energy)

FBD

(F and M balance)

At steady state velocity or ignoring inertia

Kinematic

(velocity analysis)

12

1

2

2

1

θθθθ

!!!!

NrrN

=

==

Mechanical Design

Gearheads - Spur

Inexpensive

Efficient

Low torque

Some backlash

Motor and Gearhead often combined

Mechanical Design

Gearheads - Planetary

Large reduction in a small and lightweight assembly

Large torque capacity

Considerable backlash

Identical stages can be stacked together

Mechanical Design

Gearheads - Harmonic

Flexspline has 2 fewer teeth than circular spline

One tooth advances every rotation

Huge reduction in small and lightweight assembly

Almost no backlash

Aerospace and robotics applications

HighLowMedMedHarmonic

MedHighLowHighPlanetary

LowLowHighLowSpur

CostBacklashEfficiencyTorqueType

Mechanical Design

Gearhead Comparison

Mechanical Design

Bearings (1)

Inner race

Outer race

Bearings allow for slight misalignment

Bearings are usually press fit to the shaft or housing.

Correct bearing hole diameter is critical.

Heat expansion can be used to assist assembly.

Mechanical Design

Bearings (2)Load conditions determine mounting scheme to prevent bearing “creep”. Creep is when the bearing race rolls on the mating surface.

Shaft “sees” varying load so press fit shaft

Press fit bearing race (inner or outer) that experiences cyclic loading

Housing “sees” varying load so press fit housing

Belt Drive System Centrifuge with unbalanced load

Mechanical Design

Flexible Couplings

• Coupling is used to transmit torque between two shafts.

• Coupling is rigid in torsion and flexible in bending.

• Without flexible coupling, there would be excessive loadson the shafts and bearings.

• Without flexible coupling, bearings would fail prematurelyand performance would suffer.

Angular misalignment

Parallel offset misalignment

Mechanical Design

Fixing Components to Shafts

HighLowHighSpline

MedHighMedKey

LowHighLowPin

MedLowHighClamp

LowHighLowSetscrew

CostStressTorqueMethod

unreliable

Mechanical Design

Dynamics

τθθθθθθθ =+++ )()(),()( GBCM !!!!!Ordinary differential equation:

Solution:

Numerical integration (MATLAB ODE solvers)

θτ !!!! JxmF == Dynamic forces are exerted on the mechanical components.

Dynamic forces cause deflections and stresses in the structural components.

Mechanical Design

Stress and Strain (1)

When would we not design to yield strength?

Yield Stress

Stresses in the components should never exceed the Yield Stress to prevent permanent deformation.

(elongation)

In the linear region, material acts like a spring ( F = kx ).

Mechanical Design

Heat Transfer (1)

Why are we concerned with heat transfer?

Heat transfer is energy transfer due to a temperature difference.

HOT COLD

Modes of heat transfer

Conduction Convection Radiation

SolidSolid Surface to Fluid

No Medium

Electromagnetic Waves

Mechanical Design

Heat Transfer (2)

Why is there a max continuous current for a motor?

Can we exceed this max continuous current?

Ri2

Internal heat generation

0 0.5 1 1.5 20

50

100

150

time (s )

tem

p (C

elci

us)

Motor Temp

s mall thermal inertialarge thermal inertia

Mechanical Design

Partial Differential Equations

Heat Transfer - Conduction

q” = heat flux (energy across unit area per unit time)

T = temperature

∂∂+

∂∂+

∂∂−=∇−=′′

zTk

yTj

xTikTkq

""""

Stress and Strain

Fourier’s Law

0

0

0

=+∂

∂+

∂∂+

∂∂

=+∂

∂+

∂∂

+∂

∂

=+∂

∂+∂

∂+

∂∂

zyzxzz

yyzxyy

xxzxyx

Fyxz

Fzxy

Fzyx

ττσ

ττσ

ττσ

Equations of Equilibrium

Mechanical Design

Solutions to Partial Differential Equations

Analytical solutions exist for basic geometry only:

What if the geometry is complex?

Heat Transfer – 1-D Conduction

Stress and Strain -Cantilevered beam

)3(6

2

lxEI

Pxy −=

112 )()( TLxTTxT +−=

Mechanical Design

Finite Element Method

Method to find an approximate solution to PDE by discretization.

• Solution converges as mesh is refined (Galerkin Method).

• Very time consuming.

• Computation time vs. modeling accuracy.

• What resolution mesh is required?

Stress in piston

Hook Stress

Mechanical Design

Design Study – Flexible Mirror (1)Finite Element MeshRapid positioning of laser beam

PD control with actuator saturation

High inertia (thick mirror) increases settling time.

Vibrations (thin mirror) increases settling time.

What mirror thickness minimizes settling time?

ODE+PDE

Mechanical Design

Design Study – Flexible Mirror (2)

Possible dimensional variation

with ±0.010 tolerance

How would you specify the thickness of the mirror for high performance that is robust against manufacturing tolerances?

Vibration Dominates

Inertia Dominates

Mechanical Design

Design Study – Flexible Mirror (3)

What fidelity model is required?

Computation time vs. accuracy

Mechanical Design

Design Study – Pendubot Link (1)

Motorized Joint

Free Joint

Link 2 (Balancing )

Link 1Similar to balancing a broom on the hand

* Link 1 is subject to dynamic loading conditions *

GOAL = minimize the mass of link 1 but donot exceed the yield strength of the material

Mechanical Design

Design Study – Pendubot Link (2)

* Lower Link is subject to dynamic loads *

Pendubot Balancing

Mechanical Design

Design Study – Pendubot Link (3)

•Very rough approximation of T-slot hollowing technique

• 9 Design Variables = Dia.of the holes

•Material = Aluminum 6061T-6

Simplified ModelT-slot hollowing

StressDeformationOptimal Design & Loads

Initial Design

OptimizationResults

Forces from simulation

FEM

Mesh

Mechanical Design

Design Study - Robot Link (1)

Horizontal Inverted Pendulum

Design Parameters xi :• Mass • Center of mass location• Inertia about x direction• Inertia about y direction

Optimal Mechanical Design

Optimize for

1. Controllability

2. Small unstable open loop pole

Mechanical Design

Design Study - Robot Link (1)

We can’t optimize both metrics simultaneously, so we get a family of design solutions.

Mechanical Design

Design Study - Robot Link (2)

Mechanical Design

Conclusion

• Mechanical Design and Control System Design cannot be done separately when high performance is required.

• Use the latest technologies and design tools to your advantage.

• Don’t forget about first principles and the basics.